Mathematics for the interested outsider

Lie Algebra Modules

It should be little surprise that we’re interested in concrete actions of Lie algebras on vector spaces, like we were for groups. Given a Lie algebra we define an -module to be a vector space equipped with a bilinear function — often written satisfying the relation

Of course, this is the same thing as a representation . Indeed, given a representation we can define ; given an action we can define a representation by . The above relation is exactly the statement that the bracket in corresponds to the bracket in .

Of course, the modules of a Lie algebra form a category. A homomorphism of -modules is a linear map satisfying

We automatically get the concept of a submodule — a subspace sent back into itself by each — and a quotient module. In the latter case, we can see that if is any submodule then we can define . This is well-defined, since if is any other representative of then , and , so and both represent the same element of .

Thus, every submodule can be seen as the kernel of some homomorphism: the projection . It should be clear that every homomorphism has a kernel, and a cokernel can be defined simply as the quotient of the range by the image. All we need to see that the category of -modules is abelian is to show that every epimorphism is actually a quotient, but we know this is already true for the underlying vector spaces. Since the (vector space) kernel of an -module map is an -submodule, this is also true for -modules.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.